On arc fibers of morphisms of schemes

Abstract

Given a morphism $f:X\to Y$ of schemes over a field, we prove several finiteness results about the fibers of the induced map $f_{\infty }:X_{\infty }\to Y_{\infty }$ on arc spaces. Assuming that $f$ is quasi-finite and $X$ is separated and quasi-compact, our theorem states that $f_{\infty }$ has topologically finite fibers of bounded cardinality and its restriction to $X_{\infty }\setminus R_{\infty }$, where $R$ is the ramification locus of $f$, has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers of $f_{\infty }$ when $f$ is a finite morphism of varieties over an algebraically closed field, describe the ramification locus of $f_{\infty }$, and prove a general criterion for $f_{\infty }$ to be a morphism of finite type. We apply these results to further explore the local structure of arc spaces. One application is that the local ring at a stable point of the arc space of a variety has finitely generated maximal ideal and topologically Noetherian spectrum, something that should be contrasted with the fact that these rings are not Noetherian in general; a lower bound on the dimension of these rings is also obtained. Another application gives a semicontinuity property for the embedding dimension and embedding codimension of arc spaces which extends to this setting a theorem of Lech on Noetherian local rings and translates into a semicontinuity property for Mather log discrepancies. Other applications are also discussed.